| Author |
Message |
Arnold Neumaier
science forum Guru
Joined: 24 Mar 2005
Posts: 379
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 Posted: Thu Jul 20, 2006 12:46 pm Post subject:
Re: Power Method and negative eigenvalues
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Skunk wrote:
| Quote: | Hi,
I implemented the power method algorithm in C++ in order to find the
leading eigenvalue of a real square matrix and its associated eigenvector.
My problem is that this method compute the eigenvalue with the largest
*absolute* value and that i would it to compute the largest *positive*
eigenvalue.
Is there a possiblity to adapt the power method to find the largest
positive eigenvalue ? Any ideas i can explore or should i use another
algorithm ?
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Try the Lanczsos algorithm. It is much more efficient than the power method.
Arnold Neumaier |
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Peter Spellucci
science forum Guru
Joined: 29 Apr 2005
Posts: 702
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| Posted: Thu Jul 20, 2006 11:14 am Post subject:
Re: Power Method and negative eigenvalues
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In article <44beafcc$0$12612$5402220f@news.sunrise.ch>,
Skunk <skunk@ag.com> writes:
| Quote: | Hi,
I implemented the power method algorithm in C++ in order to find the
leading eigenvalue of a real square matrix and its associated eigenvector.
My problem is that this method compute the eigenvalue with the largest
*absolute* value and that i would it to compute the largest *positive*
eigenvalue.
Is there a possiblity to adapt the power method to find the largest
positive eigenvalue ? Any ideas i can explore or should i use another
algorithm ?
Thank you very much !
|
not necessarily the best idea but simple:
with a pure power method code: compute the maxnorm of A
max_i sum_j abs(a(i,j) = N
add N to the diagonal elements of A. Now there are no negative
eigenvalues and the largest positive one is shifted by N to the right.
this might slow down convergence .
or, with inverse iteration: subtract N, use inverse iteration with shift zero
to get the reciprocal of the absolute smallest eigenvalue of the shifted
i.e. 1/(lambda_max-N)
hth
peter |
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Skunk
science forum beginner
Joined: 19 Jul 2006
Posts: 2
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| Posted: Thu Jul 20, 2006 10:29 am Post subject:
Re: Power Method and negative eigenvalues
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Roy Stogner wrote:
| Quote: | On Thu, 20 Jul 2006 00:19:11 +0200, Skunk wrote:
Is there a possiblity to adapt the power method to find the largest
positive eigenvalue ? Any ideas i can explore or should i use another
algorithm ?
The first idea that comes to mind: if the largest magnitude negative
eigenvalue of A is -g, can you use the power method to find the largest
magnitude eigenvalue of (A + gI) then subtract g to get the corresponding
eigenvalue of A?
---
Roy Stogner
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It works ! Simple and effective, many thanks !  |
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iandjmsmith@aol.com
science forum beginner
Joined: 13 Sep 2005
Posts: 15
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| Posted: Thu Jul 20, 2006 8:48 am Post subject:
Re: Power Method and negative eigenvalues
|
|
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Skunk wrote:
| Quote: | Hi,
I implemented the power method algorithm in C++ in order to find the
leading eigenvalue of a real square matrix and its associated eigenvector.
My problem is that this method compute the eigenvalue with the largest
*absolute* value and that i would it to compute the largest *positive*
eigenvalue.
Is there a possiblity to adapt the power method to find the largest
positive eigenvalue ? Any ideas i can explore or should i use another
algorithm ?
Thank you very much !
|
In
http://groups.google.co.uk/group/sci.math.num-analysis/browse_frm/thread/81322353885a6d4d
there is some discussion on finding the largest positive eigen value of
a real symmetric matrix.
Ian Smith |
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Jeremy Watts
science forum Guru Wannabe
Joined: 24 Mar 2005
Posts: 239
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| Posted: Thu Jul 20, 2006 8:47 am Post subject:
Re: Power Method and negative eigenvalues
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"Skunk" <skunk@ag.com> wrote in message
news:44beafcc$0$12612$5402220f@news.sunrise.ch...
| Quote: | Hi,
I implemented the power method algorithm in C++ in order to find the
leading eigenvalue of a real square matrix and its associated eigenvector.
My problem is that this method compute the eigenvalue with the largest
*absolute* value and that i would it to compute the largest *positive*
eigenvalue.
Is there a possiblity to adapt the power method to find the largest
positive eigenvalue ? Any ideas i can explore or should i use another
algorithm ?
Thank you very much !
|
why not just use the QR algorithm to find the whole spectrum and then search
it to find the largest positive eigenvalue? |
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Gottfried Helms
science forum Guru
Joined: 24 Mar 2005
Posts: 301
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| Posted: Thu Jul 20, 2006 7:27 am Post subject:
Re: Power Method and negative eigenvalues
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Am 20.07.2006 00:19 schrieb Skunk:
| Quote: | Hi,
I implemented the power method algorithm in C++ in order to find the
leading eigenvalue of a real square matrix and its associated eigenvector.
My problem is that this method compute the eigenvalue with the largest
*absolute* value and that i would it to compute the largest *positive*
eigenvalue.
Is there a possiblity to adapt the power method to find the largest
positive eigenvalue ? Any ideas i can explore or should i use another
algorithm ?
Thank you very much !
Would it be possible to compute the matrix-exponential EA of A |
first? Then the negative eigenvalues of A transform to the
lowest eigenvalues of EA. Then find the highest eigenvalue
of EA with the power-method and take its log.
Just sketch, no warranty, that this is efficient ...
Gottfried Helms |
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Roy Stogner
science forum beginner
Joined: 13 Jun 2005
Posts: 38
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| Posted: Thu Jul 20, 2006 3:22 am Post subject:
Re: Power Method and negative eigenvalues
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On Thu, 20 Jul 2006 00:19:11 +0200, Skunk wrote:
| Quote: | Is there a possiblity to adapt the power method to find the largest
positive eigenvalue ? Any ideas i can explore or should i use another
algorithm ?
|
The first idea that comes to mind: if the largest magnitude negative
eigenvalue of A is -g, can you use the power method to find the largest
magnitude eigenvalue of (A + gI) then subtract g to get the corresponding
eigenvalue of A?
---
Roy Stogner |
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| Back to top |
|
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Skunk
science forum beginner
Joined: 19 Jul 2006
Posts: 2
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| Posted: Wed Jul 19, 2006 10:19 pm Post subject:
Power Method and negative eigenvalues
|
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|
Hi,
I implemented the power method algorithm in C++ in order to find the
leading eigenvalue of a real square matrix and its associated eigenvector.
My problem is that this method compute the eigenvalue with the largest
*absolute* value and that i would it to compute the largest *positive*
eigenvalue.
Is there a possiblity to adapt the power method to find the largest
positive eigenvalue ? Any ideas i can explore or should i use another
algorithm ?
Thank you very much ! |
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